A063131 - OEIS

%I #14 Sep 08 2022 08:45:03

%S 55,91,215,407,493,893,1189,1343,1403,1643,1681,1961,3151,3223,3415,

%T 4063,4579,7087,7597,7979,8791,9167,10579,11227,13303,13655,14219,

%U 15487,16147,22939,23479,24341,25751,26101,27571,28757,30461,30607

%N Odd composite numbers which in base 2 contain their largest proper factor as a substring of digits.

%C The Pascal program checks to n=100000 in about a second on a 2GHz desktop, about three times as fast than the Mathematica program.

%H Amiram Eldar, <a href="/A063131/b063131.txt">Table of n, a(n) for n = 1..10000</a>

%t Do[ If[ !PrimeQ[ n ] && StringPosition[ ToString[ FromDigits[ IntegerDigits[ n, 2 ] ] ], ToString[ FromDigits[ IntegerDigits[ Divisors[ n ] [ [ -2 ] ], 2 ] ] ] ] != {}, Print[ n ] ], {n, 3, 500, 2} ]

%o (Pascal) program A063131; var n,nn,lpd:longint; nstr,dstr:string; function prime(n:longint; var d:longint):boolean; var sq,i:longint; begin{PRIME} sq := round(sqrt(n)); for i := 2 to sq do if n mod i=0 then begin d := n div i; prime := false; exit; end; prime := true; end{PRIME}; begin{MAIN} for n := 3 to 100000 do if (n mod 2=1) and (not prime(n,lpd)) then begin nn := n; nstr := ''; repeat if nn mod 2=1 then nstr := '1'+nstr else nstr := '0'+nstr; nn := nn div 2; until nn=0; dstr := ''; repeat if lpd mod 2=1 then dstr := '1'+dstr else dstr := '0'+dstr; lpd := lpd div 2; until lpd=0; if pos(dstr,nstr)>0 then write(n:8); end; end.

%o (Magma) [k:k in [3..31000 by 2] | not IsPrime(k) and IntegerToString(Seqint(Intseq(Max(Set(Divisors(k)) diff {k}),2))) in IntegerToString(Seqint(Intseq(k)),2)]; // _Marius A. Burtea_, Jan 29 2020

%o (Python)

%o from sympy import divisors, isprime

%o def ok(n):

%o     if n < 4 or n&1 == 0 or isprime(n): return False

%o     return bin(divisors(n)[-2])[2:] in bin(n)[2:]

%o print([k for k in range(10**5) if ok(k)]) # _Michael S. Branicky_, Jul 29 2022

%Y Cf. A062238, A063127.

%K base,nonn

%O 1,1

%A _Robert G. Wilson v_, Aug 08 2001

%E Extended and edited by _John W. Layman_, Apr 06 2002